"What happens in $X$ stays in $X$."Illustration: Imagine that $X$ is a little box of marbles resting on the floor of a room $Y$, and let $G$ act on $Y$ so that all the elements inside the room (including the marbles in the box $X$!) get tossed around. (So $G$ is like an earthquake, or something....) If the action of $G$ is transitive, then none of the marbles in the little box fell outside the box and onto the floor ($Y$). Even though the elements of $X$ were shuffled, they remained within $X$.